Virasoro-like Algebras And Vertex Algebras

It is well known that Virasoro algebras and vertex algebras play an important role in many branches in both mathematics and physics. The derivation Lie algebra of Laurent polynomial ring in one variable is called the Witt algebra, and the universal central extension of the Witt algebra is called the Virasoro algebra. The representations of the Virasoro algebra play a crucial role in the construction and the analysis of the structure of the integrable modules of affine Kac-Moody algebras (see [GO1] and [GO2]). Introduced by Borcherds, vertex algebras have been studied and applied in conformal field theory and string theory in theoretical physics (see [GSW]). Lattice vertex algebras form an important class in the theory of vertex algebras. These algebras play a fundamental role in constructing the moonshine module [FLM] and shaping the theory of vertex algebras.Virasoro-like algebra W, which can be considered as the generalization of (centerless) Virasoro algebra, has the basis {D(m)|m∈Z²\{0}}, where m = (m₁,m₂), with the relationThere are two ways of thinking of W as " Virasoro algebra ". First, in the palne (x,y), for m = (m₁,m₂), let D = e^{i(m₁y+m₂x)}. if we use the usual Poisson bracket {f,g} =(?) .We see that for D(m₁,m₂) = e^{i(m₂x+m₁y)} and D(n₁, n₂) = e^{i(n₂x+n₁y)}we haveThe algebra W can also be thought of as polynomial vector fields on the plane in the following way, letIt can be checked that (see [KPS]).Let L = W (?) Cd_i (?) Cd₂, where d₁,d₂ are the degree derivations. In [LT1] , the authors constructed a class of functors F_g^αfrom sl₂-modules V to L_q-modules F_g^α(V), where L_q is the skew derivation Lie algebra over the rank two quantum torus, and studied the skew derivation Lie algebra L_q-module F_g^α(V). If q = 1, L_q is the Virasoro-like algebra L, L_q-module F_g^α(V) is the L-module F^α(V). In chapter one, we study the derivations from the Virasoro-like algebra L to L-modules F^α(V), where V is a finite dimensional simple sl₂-module. At the same time, we give the first cohomology group H¹(L, F^α(V)).Diff₀R² is the algebra of area preserving diffeomorphisms of the 2-plane(see [Ba]), the basis elements of Diff₀R² isand the commutation relations are:where,From the commutation relations, we can identifyζ_fm,n with f_m,n.Note that Diff₀R² admits the following decomposition:where Diff₀⁺R² is generated by {f_m,n|m∈Z,n≥0}, H by {f_m,-1 = x^m+1|m∈Z} and Diff₀^-R² by {f_m,n|m∈Z, n < -1}. Lie algebras Diff₀⁺R² (?) H and Diff₀⁺R² are called the Lie algebras of Block type, which are the generalizations of Block algebras introduced by Block in the paper [B1]. Since these algebras are closely related to the Virasoro algebra, sometimes they are called Virasoro-like algebras. In [Sul], the author studied the quasifinite representations of the universal central extension of Diff₀⁺R² (?) H. In [Ba], Bakas studied the unitary representations of Diff₀⁺R², and discussed its significant applications in the context of quantumn fields theory. In chapter two and three, we study the Leibniz central extension of Diff₀⁺R² and its representations. We denote L_m,n = -f_m+n,n, m, n∈Z, then we getLet (?) = span_C{L_m,n|m,n∈Z,n≥0}, then (?)= Diff₀⁺R². In chapter two, we give the second Leibniz cohomology group of (?), which we find is equal to the second cohomology group of (?). Thus the universal central extension B=(?)C_Kis defined by the2-cocycleβon (?) :and thus has the following bracket:for m,m'∈Z, n,n'≥0, where k is a central element. One sees that B contains a subalgebra with basis {L_m,0,K |m∈Z}, which is isomorphic to the Virasoro algebra. In [M], it is proved that any Harish-Chandra over the Virasoro algebra is a highest weight module, a lowest weight module or a module of the intermediate series.In [KL,KS,KWY,LT2,LT3,Su1,Su2,Su3], the authors are all studied the modules with finitely dimensional weight space. In [Su1], the author studied the quasifinite representations of the universal central extension (?) of the Lie algebra of Block type (?), where (?)= span_C{L_m,n|m,n∈Z, n≥—1} with the bracketThe central extension B' is defined by the 2-cocycleand thus the bracket:The author proved that a quasifinite irreducible B'-module is a highest or lowest weight module. In chapter three, we study the subalgebra (?) of Block type Lie algebra (?), but they have different universal central extension, we will see that the representation theory for the Lie algebra B is different from that for B' . In the vertex algebras part, we study vertex algebras and their modules associated with possibly degenerate even lattices from the point of view of vertex algebra extensions. Several known results are recovered and a number of new results are obtained. We also study modules for Heisenberg algebras and we classify irreducible modules satisfying certain conditions and obtain a complete reducibility theorem.